Here’s a big number:
Try different single-digit divisors. No remainders.
This is an example of purposeful practice – exposing the wonder of mathematics whilst providing a reason to practise lots of of bus stop division.
You might want to start by asking pupils to come up with their own dividend “in the tens of millions” and try different divisors. (Here for a quick primer on the mathematical language.) Inevitably they will end up with remainders, which they may or may not carry into decimal places. Then let show them this “magic” number.
Questions to ask:
- What divisors does this work for and why? (Purposeful practice)
- What other dividends could I make like this? (Purposeful practice + reasoning)
- What smaller dividends could I make like this? (reasoning)
- What is the smallest dividend I could make that all numbers 1-9 will divide into without remainders? (reasoning)
Whilst I would want everyone in the class to understand the reasoning through a whole-class discussion, you may have some learners who need the practice on bus stop long division and spend most of their time doing this. Those that are confident with this technique can spend their time exploring deeper into the structure of the number.
Whilst we are on the subject of “Bus Stop”, maybe this technique actually has nothing to do with standing in line waiting for a bus:
On Saturday, I presented at #mixedattainmentmaths (Powerpoint is here), the first in hopefully a series of conferences bringing together teachers and educationalists to share ideas and experiences of teaching in a mixed environment. As I had expected, most of the presenters had a deeply held conviction in the validity of organising our classes in this way, whether it be for social justice reasons or due to educational research demonstrating that the overall progress of learners is improved compared to a setted environment.
I admire those who are championing this cause. Since starting teaching I have felt a sense of unease about putting children into sets, but the pragmatist in me can see the reasons why it is the predominant way of organising secondary maths in the UK.
My talk was mostly about some of the resources that we have developed over the last year or so of teaching our Year 7 and Year 8 classes in mixed attainment groupings (currently with a small nurture group in Year 7). If you would like to read more about some of the changes that we have made, Gwen Tresidder from NCETM wrote a case study on our school here.
Before looking at some maths, I did touch on some of the considerations for schools moving to mixed. It is a significant change to make to a department and one that needs to be considered carefully. In particular:
- Is there a problem with classroom culture in your lower sets or indeed sometimes top sets where maths can be seen as something that is all about speed and getting the right answer rather than reasoning and problem solving?
- How will teachers be supported? I believe that we have been successful so far in moving to mixed groups and the key to this has been our weekly 1 hour collaborative planning sessions. e. 1 hour for Year 7 teachers and 1 hour for Year 8.
- How to plan the low threshold high ceiling activities that provide suitable challenge for a wide range of learners. The “Toy Story” tasks, I called them, where different learners can be working at different levels at the same time in a manageable way.
- How to develop teachers’ skills in questioning a mixed attainment group. I have noticed that there is more whole-class dialogue in our mixed attainment classes, but it requires skill from the teachers in directing these questions and managing the responses so that everyone learns something from the explanations given. This recent blog from Dani Quinn has oodles of great advice on this.
- The need to develop children’s skills and habits in providing careful explanations of their mathematical reasoning. We ask them to stand up to make a contribution, for example. Initially I was uneasy about this, but I have really seen the benefit once it becomes established. You can see children carefully constructing their explanation in their head before standing up. When they are ready, they stand up and usually give a coherent explanation using correct terminology. It makes them slow down, think first, and I believe they know it is a better contribution and feel proud as a result.
- How to convince parents that their child is being given “appropriate” work to do. This has been particularly challenging with the higher attainers, especially when that child is in top sets for other subjects. Some parents feel we are just following a fad. We have made efforts to explain the reasons behind what we are doing, but we probably need to do more.
The focus of this workshop was on that last point, i.e. how to provide challenge to higher attainers in mixed groups. I will be writing more about that over the coming weeks, so please follow this blog if you are interested.
UPDATED POST. I used this task at my workshop at #mixedattainmentmaths on Saturday. I asked all teachers to have a go at this task but to do it in what they thought was the most obvious / simplest way. An interesting experiment: what is obvious to some is not to others. Of the solutions that I managed to take in, these were the choices:
This looks like a very useful open-ended task which provides an opportunity for creative solutions and rich discussion. I have produce some printouts here.
In my view, the value in this activity is in representing each area as a fraction calculation.
According the Australian blog where I first read about this task, this is one of the most common first solutions
I’d be looking for some rationalising as to why the red area is a quarter. For example:
There are 100 solutions posted here!
And on a Prezi here enabling you to zoom into each one individually.
This is potentially very high ceiling. If students are struggling to come up with suitably challenging solutions of their own, you could always ask:
Show why this is a quarter:
Have a go first yourself. I think this is a pretty mammoth task! This one caught my eye, but you might want to have a look at the 100 solutions to find something a bit easier!
Since this post was originally written back in January, I have used this task a couple of times at conferences and had some really good discussions on this example. If you want a rather big hint, scroll down to see an animation. Or you can find the Geogebra file here.
As I write this, around 35 teachers from Shanghai are heading back home after 2 weeks teaching Maths in our primary schools. In our Maths Hub, around 250 teachers took time out of their classrooms to watch and unpick the mathematical content of a KS1 or KS2 lesson. Combined with a similar-sized group that came in December, maybe 6000-7000 teachers in England have had this opportunity. It’s been a significant undertaking. Now is a good time to reflect on where we go next and ask the question – was it worth it?
Very good feedback was received from the teachers that came to watch our lessons. Maybe the audience was self-selecting. Maybe those teachers who have years of experience and have “seen it all before” stayed away. Or those who have a strong philosophy about how we should teach children maths which conflicts with their perception of what goes on in Shanghai classrooms didn’t fancy it. But those who came with an open mind and a sensible level of expectation were inspired and took something away. We didn’t see perfect lessons. Language was sometimes an issue but less than you might imagine. The Chinese teachers were not used to teaching the wide range of attainment we see in our classrooms. There was no differentiation in the lessons. There were usually plenty of adults on hand to help the children, something which is clearly unrealistic in the normal run of things.
What we did see were some carefully constructed lessons from practitioners who focus entirely on Maths teaching. In a system that asks its teachers to teach 2-3 x 35 minute lessons per day with plenty of time for professional development, that regularly has 10-15 teachers observing and unpicking lessons, it is not surprising that these teachers know their stuff! We saw the micro-progression building up a solid understanding of underlying concepts. We saw the experience of a mastery approach that has been in place for many years and has percolated down to thousands of teachers. For an excellent description of some of the techniques used read Tim Brogan’s post here.
So a good experience for those involved. But that doesn’t answer the question “was it worth it?” I don’t know the full cost of the exchange. I guess I could do an FoI request to find out, but I’m estimating £3000 x 75, so about £225,000 for the trip. But the bigger cost was 70 odd teachers being out of their classrooms for 2 weeks. It seems so hard to carve out enough time to see other teachers in our own schools, let alone travel half way across the world to watch excellent practitioners. In these times of budget cuts we should be grateful that there is this investment into Maths education. And it has given a real impetus to CPD for Maths. But I can’t help thinking that an alternative could be 700 teachers spending a day observing excellent practitioners in their local area. Or several thousand teachers having an extra hour to observe an excellent lesson in their own school. Would this have a greater impact on more children?
The good news is that this sort of “learning from each other” between schools is available through the Maths Hubs Teaching for Mastery programme. This provides schools (Primary and Secondary) with the opportunity to work with up to 6 other schools in their local area in a TRG – a group that meets regularly throughout the year to observe and unpick each others’ lessons as well as working together on lesson design and curriculum planning. The Maths leader from one of these schools will have received a year of training from NCETM and will be the “Mastery Specialist” but in reality this should be seen as a genuine collaboration between practitioners. The schools involved receive a reasonable level of funding via the Maths Hubs to cover the time the teachers are out of lessons. To find out more, contact your local Maths Hub.
We have gained a lot from this exchange, but I wonder if we are hitting diminishing returns to go for a fourth year in a row. There are now enough people around the country who have experience of what happens in Shanghai classrooms. The focus should be squarely on helping our teachers adapt what we have seen for the English classroom. To take the best bits and meld them into currently existing good practice. The need now is to provide more teachers with the time to undertake high quality collaborative lesson design and to learn from each other by observing each other. This is what really makes the difference in Shanghai and this is what will make the difference here.
No 4 in a series of posts based on Thinking Mathematically (1985) by Mason, Burton, Stacey
This is a classic task for working systematically:
This is a classic question that I usually use when teaching loci and constructions. I usually scaffold it with a diagram, but I like the approach here that promotes students to introduce their own diagram as a way of getting closer to the problem. Next time I use this I will make sure I give it plenty of time, get students discussing the problem in pairs and really resist providing a diagram and thus removing some of the joy of discovering it!
A goat is tethered by a 6 metre rope to the outside corner of a shed measuring 4 metres by 5 metres in a grassy field. What area of grass can the goat graze?
If I felt the need to make this question easier, I might start with the rope being 4 metres long, then try 5, then try 6. In fact, this could be taken a lot further, what happens as the length of rope is increased further? Try 10m, 50m, 100m. What happens when the rope gets really long?
Or, as a way in to this problem I might ask:
A goat is tethered by a 6 metre rope to a simple post in an open field. What is the area of grass it can eat?
I will have this question ready written, but hidden on the same slide on the board so that I can show it to the whole class if I feel they need it. But alternatively I might just suggest this to individual students if they are really struggling and making no progress. Either way, the point is that it is not another question that I want them to work on, it is a problem solving strategy that I am exposing them to: If the problem seems too difficult, try to find a simpler scenario as a way in to the problem. Solve that, and then think about what knowledge and methods you used and how they could be applied to the main problem.
Some great questions here from @solvemymaths. Depending on the class, I might present 4 of these and then get them to try creating their own using the idea of regular polygons. Regular polygons seem to be quite popular on the new GCSE.
All these puzzles are constructed using regular shapes.
Source: What’s the missing angle?
No 3 in a series of posts based on Thinking Mathematically (1985) by Mason, Burton, Stacey
This is a classic task for working systematically:
It was once claimed that there are 204 squares on an ordinary chessboard.
Can you justify this claim?
I like this way of stating the problem, rather than just “how many squares on a chessboard?” because it gets straight to the nub of the issue – we are looking at different sized squares, some of which overlap.
No 2 in a series of posts based on Thinking Mathematically (1985) by Mason, Burton, Stacey
This is the sort of exercise I can envisage taking a number of different paths depending on what my students do with it which is exciting. The book walks through a generalisation by looking for the lowest 4-digit palindromic number, 1001 and then noting that subsequent palindromic numbers can be found by adding 110. Since 1001 and 110 are multiples of 11, then all numbers in this series are multiples of 11. However, this series misses out other palindromic numbers, e.g. 7557 so we need to refine it further.
I am intrigued to see if this is indeed a path my students would follow or if we would discover something else in these numbers. Depending on the class, I might start by asking “how many 4 digit palindromic numbers are there?” Before getting into the general, I would see this as an opportunity for purposeful practice of long division if that was something that my students require. Some students might need a fair amount of direction to reach a proof, but I would aim to make sure that all students left this lesson with an appreciation of that proof even if I had to lead them through it.
No 1 in a series of posts based on Thinking Mathematically (1985) by Mason, Burton, Stacey
This question prompts learners to come up with their own specific example(s) of a price to apply the discount and sales tax to. It made me think that there are many questions I use in class which I could adapt (i.e. remove the specific example) to get my students better acquainted with this step.
I used £100 and performed the calculation mentally. It prompted me to reflect on how I chose £100 and how I would help my students make good choices for examples.
I think a number of my students might think that they have seen something a bit like this before:
"A price is reduced by 20% in a sale but this price is then reduced by a further 10% by using a voucher code"
Amy says that the overall reduction is 30%. Explain why she is wrong.
Depending on the class, I might use these two examples together to highlight the process of moving through from the specific by choosing good examples to the general using basic algebraic notation with decimal multipliers.
In the first example, we need to get to:
0.8 x 1.15 x P = 1.15 x 0.8 x P
In the second case:
0.8 x 0.9 x P ≠ 0.7 x P
These statements alone are insufficient “explanations” in the context of the question, so there could be a good opportunity here to work on some specific exam technique as these types of questions often pose difficulty for students.
I spend time on Twitter and reading blogs. I “flit” from one thing to the next not spending much time engaging with any one “thing”. Usually this is enjoyable, and feels somewhat productive, but I also feel the need to engage more fully and deeper with a single idea to achieve a greater sense of accomplishment. I have bought various books over the last few months, mostly second hand. I have consciously been trying to spend less time on Twitter and more time reading.
The advice from a few responses to this tweet was to start with Thinking Mathematically by Mason, Burton and Stacey, 1985
I have had the privilege of participating in a two separate sessions run by John Mason and Anne Watson recently and I have read various articles of theirs so I had a sense of what the book would be like. I am now part way through it and finding it stimulating.
This is a book that requires active engagement, not just passive reading. It’s a book that presents a need to grab some paper and a pencil on occasions; in other instances, I have challenged myself to do the exercises mentally.
It has also stimulated an intention to write this series of blog posts. As with everything on this blog, my primary objective is for my own reflection and categorisation of ideas such that I might find them again one day for use in planning lessons. Hopefully a happy side-effect is that it stimulates others’ thinking and maybe motivates one to try something and write about it.
Here is my list of my posts so far, which I will increase at I write more.
- Chessboard Squares
- Tethered Goat
- Cubes Cubed